MATH 313 Midterm: MATH 313 2004 Winter Test 2
Document Summary
Be sure that this examination has 10 pages, including this cover. Special instructions: no calculators, books, notes, or other aids allowed, answer all 8 questions. All questions are worth 5 marks: give your answer in the space provided. If you need extra space, use the back of the page: show enough of your work to justify your answer. Problem 1: find all primes p for which the congruence x 2 + 3x + 1 0 (mod p) has a solution. Problem 2: show that there are in nitely many primes p of the form p = 4k + 1. Find the rst 5 convergents of the continued fraction expansion for e = [2, 1, 2, 1, 1, 4, 1 . If d > 1, show that the continued fraction expansion of d2 1 is given. Problem 4: by [d 1, 1, 2d 2, 1, 2d 2 . (the string 1, 2d 2 repeats).
